Following Ref. [1]}, we introduce the expansionNote that we performed an \(s \leftrightarrow t\)  transformation compared to Eq. (REF ).
\(f(t,u)= f(t,-s-t) = \left( \frac{\kappa \hbox{}}{2} \right)^2 {1 \over s t u} + |\beta _{R^3}|^2 {t u \over s} - |\beta _\phi |^2 {1 \over s} + \sum _{k \ge j \ge 0 } a_{k,j} s^{k-j} t^j \, .\) 
